where \(\hat{\sigma }_v^2\)  is an estimator of the variance \(\sigma _v^2\) . One of the ways to estimate \(\sigma _v^2\)  is the estimator \(\hat{\sigma }_v^2\)  based on the method of moments, as originally proposed by [1]}. For this estimator, approximately unbiased estimators of MSEs of (REF ) were derived in [2]}:
\(\begin{split}\operatorname{mse}(\hat{\theta }_i^{\mathrm {FH}})= \hat{\gamma }_i\hat{\psi }_i + (1-\hat{\gamma }_i)^2 & \Biggl [ \mathbf {z}_i^{\prime } \Biggl ( \sum _{j=1}^M \frac{\mathbf {z}_j\mathbf {z}_j^{\prime }}{\hat{\psi }_j+\hat{\sigma }_v^2}\Biggr )^{-1} \mathbf {z}_i + \frac{4M}{\hat{\psi }_i+\hat{\sigma }_v^2} \Biggl ( \sum _{j=1}^M \frac{1}{\hat{\psi }_j+\hat{\sigma }_v^2}\Biggr )^{-2} \Biggr . \\& \Biggl . -2\hat{\sigma }_v^2 \Biggl ( \sum _{j=1}^M \hat{\gamma }_j \Biggr )^{-3} \Biggl \lbrace  M\sum _{j=1}^M \hat{\gamma }_j^2 - \Biggl ( \sum _{j=1}^M \hat{\gamma }_j \Biggr )^2 \Biggr \rbrace  \Biggr ] , \qquad i=1,\ldots , M.\end{split}\) 
